# Rate of convergence precise meaning/definition

In the context of mathematical statistics, people often say MLE's rate of convergence is $$\sqrt{n}$$ as $$\sqrt{n}(\hat{\theta}_n-\theta_0) \to N(0,V)$$ in distribution. But is there any official definition of rate of convergence--is it simply the $$f(n)$$ that you multiply to the diminishing sequence of random variable (meaning that it goes to 0 in probability) so that the product does not degenerate or blow up (to infinity)? Is there a proof that such $$f(n)$$ is unique up to multiplication/addition of a constant?

Suppose $$f(n) X_n$$ converges in distribution.
If $$g(n)/f(n) \to 0$$ then $$g(n) X_n = \frac{g(n)}{f(n)} f(n) X_n$$ converges in distribution to $$0$$ (e.g., by Slutsky's theorem). Similarly if $$g(n) / f(n) \to \infty$$ then I think you can show $$g(n) X_n$$ blows up and does not converge in distribution, although I may be wrong about this.